Fungrim entry: f79ff0

\frac{z {e}^{x z}}{{e}^{z} - 1} = \sum_{n=0}^{\infty} B_{n}\!\left(x\right) \frac{{z}^{n}}{n !}

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 2 \pi \;\mathbin{\operatorname{and}}\; z \ne 0

TeX:

\frac{z {e}^{x z}}{{e}^{z} - 1} = \sum_{n=0}^{\infty} B_{n}\!\left(x\right) \frac{{z}^{n}}{n !}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 2 \pi \;\mathbin{\operatorname{and}}\; z \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`Sum`	$\sum_{n} f(n)$	Sum
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("f79ff0"),
    Formula(Equal(Div(Mul(z, Exp(Mul(x, z))), Sub(Exp(z), 1)), Sum(Mul(BernoulliPolynomial(n, x), Div(Pow(z, n), Factorial(n))), For(n, 0, Infinity)))),
    Variables(z, x),
    Assumptions(And(Element(x, CC), Element(z, CC), Less(Abs(z), Mul(2, Pi)), NotEqual(z, 0))))

Topics using this entry

Bernoulli numbers and polynomials